This invention relates to an improvement of a method for growing a single crystal of compound semiconductor.
Compound semiconductor has various kind of compositions--GaAs, InP, InAs, GaP, InSb, etc. Among the components the element of group V has a high dissociation pressure. It is difficult to grow a stoichiometric single crystal because of this high dissociation pressure of the element of group V.
An LEC (Liquid Encapsulated Czochralski) method is one of most preferable methods for growing a single crystal of compound semiconductor. The LEC method has advantages that circular wafers are easily obtained and semi-insulating GaAs single crystal is grown without doping Cr etc. It is an excellent method from the industrial viewpoint.
The LEC method has many improved variations. However there are some inherent disadvantages in conventional LEC methods.
In all LEC methods a single crystal is pulled up using a seed crystal from a melt of compound semiconductor covered with a liquid encapsulant under an inactive gas pressurized at several atm to several tens atm to prevent the element of group V from escaping.
When the single crystal emerges from the liquid incapsulant, it is rapidly cooled by a vehement convection of inactive gas above the liquid encapsulant. Strong thermal stresses occur in the rapidly cooled crystal, which result in many lattice defects in the crystal.
An ingot grown by the LEC method is sliced into many thin wafers. The wafers are etched. Then etch pit density (EPD) is measured. The EPD of the wafers sliced from the LEC-grown crystal is about 10,000/cm.sup.2 to 100,000/cm.sup.2.
Many improvements of LEC methods have been proposed to overcome the disadvantages of conventional LEC methods.
One proposal is an idea of preventing the generation of strong temperature gradients in a cooling single crystal. For this purpose the cooling by the convection of the inactive gas must be suppressed.
To suppress the cooling, the thick liquid encapsulant seems to be effective. Namely instead in gas a single crystal is slowly cooled in a thick liquid encapsulant instead of an inactive gas. The liquid encapsulant plays a role as heat-insulator. However this method consumes great amount of liquid encapsulant. Because the single crystal rotates in the liquid encapsulant with a high viscosity, it is difficult to control the diameter of single crystal. Although this proposal is able to be put into practice, it has these disadvantages.
Another proposal is to alleviate the temperature gradients by using plural heaters. This improvement may be called "multi-heater method". More than two heaters and heat-insulators are installed along a vertical direction to alleviate the vertical temperature gradient. Rising in a moderate temperature gradient, a single crystal is slowly cooled. The purpose of the improvement is to heighten the thermal uniformity in the space above the liquid encapsulant.
Third proposal is an idea of doping isoelectronic impurities. Isoelectronic impurity is one which has the electronic property same as the component elements of crystal. In the case of III-V compound semiconductor, isoelectronic impurity is one of the element of groups III and V other than the components of the base compound.
For example, In, B, Sb, Al, P etc., are isoelectronic impurities for GaAs.
If some isoelectronic impurities are doped with more than 10.sup.18 atoms/cm.sup.3, the EPD of the doped crystal is reduced to a great extent. Doping of other isoelectronic impurities are of no use.
For example for GaAs single crystal, the isoelectronic impurities In, Sb and B are effective to reduce the EPD.
However the reason why the isoelectronic impurities reduce EPD has not been clearly explained yet.
Distribution coefficient is defined as a quotient of an impurity concentration in solid divided by an impurity concentration in liquid or melt when the solid phase and the liquid phase are in equilibrium.
For example, the distributiion coefficient of In in GaAs is 0.1 to 0.15.
To dope In to a crystal, the GaAs melt from which the crystal is pulled must include In at six to ten times as much as the prescribed concentration in a crystal.
While a single crystal is pulled up from a melt including a high impurity concentration, the impurity concentration in the melt is increasing, if the distribution coefficient of the impurity is less than 1.
If In concentration is 5.times.10.sup.18 /cm.sup.3 at the front portion of a GaAs crystal grown by an LEC method, In concentration would attain to higher values at middle portion or back portion of the crystal.
High impurity concentration brings about impurity precipitation. The portions including impurity precipitation cannot be used as substrates on which semiconductor devices are fabricated.
The improved 1EC method to dope an impurity to reduce EPD must face with the new difficulty-impurity precipitation.
If the impurity concentration in melt is high, impurity precipitation appears at an early stage. If the impurity concentration is low, EPD can not be reduced enough, although the beginning of precipitation delays.
Why does the impurity precipitation occur? This has not been solved yet.
The beginning of impurity precipitation depends not only on the impurity concentration in melt but also the pulling speed. Pulling speed is defined as a vertical speed of an upper shaft. The seed crystal as well as the growing crystal are pulled by the upper shaft. Then the pulling speed is equivalent to the growing speed in case of LEC methods.
In general the smaller pulling speed delays more the beginning of impurity precipitation.
The multi-heater method, one of improved LEC methods, cools a growing crystal in a quasi-uniform thermal environment with a very low temperature gradient to reduce thermal stress. However this improvement seems to advance the beginning of impurity precipitation.
Two improvements of LEC methods to reduce EPD--doping of isoelectronic impurity and cooling in low temperature gradient--make the impurity precipitation problem more serious.
As mentioned before, low pulling speed delays the beginning of precipitation. However if a single crystal was pulled up at a constant low speed, the crystal growth would take much long time. It is undesirable from an economical viewpoint.
Considering the fact that the beginning of precipitation depends not only on the impurity concentration but also on the temperature gradient and the pulling speed, this inventors suspected that the cause of impurity precipitation might be "supercooling".
When a liquid is cooled, it becomes a solid at a freezing point (melting point) in general. However if cooling rate is low enough, the liquid state is kept below the freezing point. This phenomenon is called supercooling.
The condition for occurrence of supercooling in a Czochralski method has been already considered.
The condition for occurrence of supercooling is given by EQU G.sub.l /R&lt;(G.sub.l /R).sub.c K.sub.e ( 1)
where K.sub.e is an effective distribution coefficient, G.sub.l is a temperature gradient in melt near a solid-liquid interface, and R is a growth rate (i.e. a pulling speed). The round bracket suffixed with c signifies a critical value of the bracket.
The critical value of (G.sub.l /R) is given by ##EQU1## where k is an equilibrium distribution coefficient, .DELTA.T is a decrement of melting point due to the impurity inclusion in melt and D is a diffusion constant.
The impurity concentration c varies according to the equation EQU c=c.sub.0 (1-g).sup.k-1 ( 3)
where k is a distribution coefficient, g is a fraction solidified and c.sub.0 is an initial impurity concentration in melt.
The effective distribution coefficient K.sub.e is given by ##EQU2## where .delta. is a thickness of solute boundary layer. In equilibrium the thickness .delta. is zero. Although the rising single crystal is rotated to equalize the thermal environment near the solid-liquid interface, the melt and solid near the interface have transient thermal fluctuations.
In the case of Czochralski method the thickness is given by EQU .delta.(CZ)=1.6D.sup.1/3 .nu..sup.1/6 .omega..sup.-1/2 ( 5)
where .nu. is a dynamical viscosity of melt and .omega. is a relative angular velocity between the crystal and the crucible.
The decrement of the melting point is in proportion to the impurity concentration c in the melt. Then .DELTA.T is written as EQU .DELTA.T=mc (6)
where m is a constant multiplier.
From Inequality (1), Eq. (2), Eq. (3) and Eq. (6), the condition for occurrence of supercooling is given by ##EQU3##
Hereafter the condition for occurrence of supercooling will be called as "supercooling condition" and the condition for non-occurrence of supercooling will be called as "non-supercooling condition" for simplicity.
If we assume R.delta./D is much less enough than unity, the effective distribution coefficient K.sub.e is nearly equal to the equilibrium distribution coefficient k. Using this approximation, Eq. (7) can be written as ##EQU4##
In the case of LEC-growth of GaAs EQU .nu..apprxeq.4.times.10.sup.-3 cm.sup.2 /sec EQU D.apprxeq.1.times.10.sup.-4 cm.sup.2 /sec
The rotation speed .omega. depends on various factors. If we assume the rotation speed of the upper shaft is +2 RPM and that of the lower shaft is -20 RPM, The relative rotation speed becomes 22 RPM. Then the angular velocity .omega. is calculated as EQU .omega.=2.3/sec
Substituting these values into Eq. (5), we obtain EQU .delta.=0.02 cm
If the pulling speed is 10 mm/H, the value of R.delta./D is roughly estimated as EQU R.delta./D.apprxeq.0.05
This is much less than unity. Therefore in the ordinary LEC method the effective distribution coefficient K.sub.e can be replaced by the equilibrium distribution coefficient k in Eq. (4).
Then Inequality (8) is a supercooling condition.
The decrement of the melting point can be expressed in terms of gram equivalence N, because the gram equivalence N is in proportion to the concentration.
In the case of GaAs melt, EQU T=360N (9)
This relation is independent of the kind of impurity. Here we will consider the GaAs melt doped with In. The weight percent of In in GaAs melt is denoted by "w". The average atomic weight of GaAs is 72.3. The atomic weight of In is 114.82. Then the gram equivalence N is written as EQU N=0.63 w (10)
From Eq. (9) and Eq. (10), .DELTA.T becomes EQU .DELTA.T=227 w (11)
The weight percent w is given by an equation similar to Eq. (3), in which c.sub.0 is replaced by an initial gram equivalence w.sub.0.
If we take w.sub.0 and G.sub.l as explicit parameters, the supercooling condition is written as ##EQU5## where ##EQU6##
The equilibrium distribution coefficient k of In in GaAs melt is approximately 0.1. The diffusion constant D was already given for In in GaAs melt. Using these values of k and D, the constant Q is calculated as EQU Q=4.88.times.10.sup.-7 (cm.sup.2 /sec.degree.C.)
If we assume G.sub.l =50.degree. C./cm and w.sub.0 =0.015 (1.5 wt%), the non-supercooling condition is given by EQU R.ltoreq.1.62.times.10.sup.-3 (1-g).sup.0.9 cm/sec (14)
For the pulling speed R, mm/H is more practical unit than cm/sec. In the practical unit of mm/H for R, the above condition is written as EQU R.ltoreq.58(1-g).sup.0.9 mm/H (15)
Till now we will consider the supercooling condition with a typical example. Inequality (15) shows that the upper limit of pulling speed R is determined by the fraction solidified g.
Although the fraction solidified has a definite physical meaning, g is not always an observable variable. Then under some assumption optimum change of the pulling speed R will be considered.
From Inequality (12), the non-supercooling condition is given by EQU R.ltoreq.QG.sub.l w.sub.0 (1-g).sup.1-k ( 16)
Here the relation between the pulling time and the fraction solidified will be calculated.
The sectional area of a pulled single crystal, the density of the crystal and the initial weight of material melt are denoted by S, .rho. and W respectively. If the single crystal is pulled up at a constant speed R.sub.0, the full length would be W/.rho.S at g=1, in which all melt is solidified. The time required for pulling the single crystal is given by W/.rho.SR.sub.0.
Instead of a constant speed, we assume that the pulling speed R is changed as a function of fraction solidified. For example, we assume the following equation EQU R=R.sub.0 (1-g).sup.1-h ( 17)
where h is a constant of 0 to 1, and R.sub.0 is an initial pulling speed.
The fraction solidified g satisfied the equation ##EQU7##
Differentiating Eq. (18) and substituting Eq. (17), we obtain a differential equation ##EQU8##
The solution is ##EQU9##
As mentioned before, h is a constant of 0 to 1. If h is equal to the upper limit 1(h=1), Eq. (20) would coincide with the ordinary pulling with a constant speed R.sub.0.
As the value h becomes smaller, the total time for pulling up the single crystal increases in inverse proportion to h.
Substituting Eq. (20) into Eq. (17), we obtain the pulling speed R as a function of time t, ##EQU10##